## What do we mean by 'Equivalent Projective representation"?

I know that we say two representations R and R' of a group G is equivalent if there exists a unitary matrix U such that URU^(-1)=R'.
But what do we mean by equivalent projective rerpesentations?
I've heard of the theorem that the SO(3) group has only 2 inequivalent projective representations. But what does that exactly mean?
I am very interested in projective representation because it's projective representation rather than ordinary representation that represents symmetry in Quantum Mechanics since the vector A and exp(id)A represent the same physical state.
So does anyone know if there are some books that can serve as an introduction to projective representations?
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 Mentor Since you haven't received any replies, I will mention that "Geometry of quantum theory" by Varadarajan covers projective representations and their relevance to quantum mechanics. I hesitate to recommend it because I find it very hard to read, but I don't know a better option.
 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus I've never done anything about projective representations before, so this post is just a guess. But it would make sense to define first $$Z=\{cI_n~\vert~c\in \mathbb{R}\}$$ Then we define a projective representation as a group homomorphism $$\rho: G\rightarrow GL_n(\mathbb{R})/Z$$ This last group is often called $PGL_n(\mathbb{R})$, or the projective general linear group. Given, $\rho,\rho^\prime$ projective representations, it would make sense to define them equivalent if there exist $U\in O_n(\mathbb{R})/Z$ such that $$\rho(g)=U\cdot \rho^\prime(g)\cdot U^{-1}$$ for all $g\in G$. The complex case is similar.